Theorem sibfima 34502

Description: Any preimage of a singleton by a simple function is measurable. (Contributed by Thierry Arnoux, 19-Feb-2018.)

Hypotheses

Ref	Expression
sitgval.b	⊢ 𝐵 = (Base‘𝑊)
sitgval.j	⊢ 𝐽 = (TopOpen‘𝑊)
sitgval.s	⊢ 𝑆 = (sigaGen‘𝐽)
sitgval.0	⊢ 0 = (0g‘𝑊)
sitgval.x	⊢ · = ( ·𝑠 ‘𝑊)
sitgval.h	⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))
sitgval.1	⊢ (𝜑 → 𝑊 ∈ 𝑉)
sitgval.2	⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)
sibfmbl.1	⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀))

Assertion

Ref	Expression
sibfima	⊢ ((𝜑 ∧ 𝐴 ∈ (ran 𝐹 ∖ { 0 })) → (𝑀‘(◡𝐹 “ {𝐴})) ∈ (0[,)+∞))

Proof of Theorem sibfima

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sibfmbl.1	. . . 4 ⊢ (𝜑 → 𝐹 ∈ dom (𝑊sitg𝑀))
2		sitgval.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑊)
3		sitgval.j	. . . . 5 ⊢ 𝐽 = (TopOpen‘𝑊)
4		sitgval.s	. . . . 5 ⊢ 𝑆 = (sigaGen‘𝐽)
5		sitgval.0	. . . . 5 ⊢ 0 = (0g‘𝑊)
6		sitgval.x	. . . . 5 ⊢ · = ( ·𝑠 ‘𝑊)
7		sitgval.h	. . . . 5 ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊))
8		sitgval.1	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ 𝑉)
9		sitgval.2	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures)
10	2, 3, 4, 5, 6, 7, 8, 9	issibf 34497	. . . 4 ⊢ (𝜑 → (𝐹 ∈ dom (𝑊sitg𝑀) ↔ (𝐹 ∈ (dom 𝑀MblFnM𝑆) ∧ ran 𝐹 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(◡𝐹 “ {𝑥})) ∈ (0[,)+∞))))
11	1, 10	mpbid 232	. . 3 ⊢ (𝜑 → (𝐹 ∈ (dom 𝑀MblFnM𝑆) ∧ ran 𝐹 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(◡𝐹 “ {𝑥})) ∈ (0[,)+∞)))
12	11	simp3d 1145	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(◡𝐹 “ {𝑥})) ∈ (0[,)+∞))
13		sneq 4578	. . . . . 6 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
14	13	imaeq2d 6021	. . . . 5 ⊢ (𝑥 = 𝐴 → (◡𝐹 “ {𝑥}) = (◡𝐹 “ {𝐴}))
15	14	fveq2d 6840	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑀‘(◡𝐹 “ {𝑥})) = (𝑀‘(◡𝐹 “ {𝐴})))
16	15	eleq1d 2822	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑀‘(◡𝐹 “ {𝑥})) ∈ (0[,)+∞) ↔ (𝑀‘(◡𝐹 “ {𝐴})) ∈ (0[,)+∞)))
17	16	rspcv 3561	. 2 ⊢ (𝐴 ∈ (ran 𝐹 ∖ { 0 }) → (∀𝑥 ∈ (ran 𝐹 ∖ { 0 })(𝑀‘(◡𝐹 “ {𝑥})) ∈ (0[,)+∞) → (𝑀‘(◡𝐹 “ {𝐴})) ∈ (0[,)+∞)))
18	12, 17	mpan9 506	1 ⊢ ((𝜑 ∧ 𝐴 ∈ (ran 𝐹 ∖ { 0 })) → (𝑀‘(◡𝐹 “ {𝐴})) ∈ (0[,)+∞))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052   ∖ cdif 3887  {csn 4568  ∪ cuni 4851  ^◡ccnv 5625  dom cdm 5626  ran crn 5627   “ cima 5629  ‘cfv 6494  (class class class)co 7362  Fincfn 8888  0cc0 11033  +∞cpnf 11171  [,)cico 13295  Basecbs 17174  Scalarcsca 17218   ·_𝑠 cvsca 17219  TopOpenctopn 17379  0_gc0g 17397  ℝHomcrrh 34157  sigaGencsigagen 34302  measurescmeas 34359  MblFnMcmbfm 34413  sitgcsitg 34493

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5521  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-ov 7365  df-oprab 7366  df-mpo 7367  df-sitg 34494

This theorem is referenced by:  sibfinima  34503  sitgfval  34505  sitgclg  34506